Difference between revisions of "User:Tohline/SSC/Synopsis StyleSheet"
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to find one or more radially dependent, radial-displacement eigenvectors, <math>~x \equiv \delta r/r</math>, along with (the square of) the corresponding oscillation eigenfrequency, <math>~\omega^2</math>. | to find one or more radially dependent, radial-displacement eigenvectors, <math>~x \equiv \delta r/r</math>, along with (the square of) the corresponding oscillation eigenfrequency, <math>~\omega^2</math>. | ||
! style="vertical-align:top; text-align:left;" rowspan=" | ! style="vertical-align:top; text-align:left;" rowspan="5"| | ||
The second derivative of the free-energy function is, | The second derivative of the free-energy function is, | ||
<table border="0" cellpadding="5" align="center"> | <table border="0" cellpadding="5" align="center"> | ||
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! style="text-align:center;" width="50%" |<b>Variational Principle</b> | |||
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! style="vertical-align:top; text-align:left;" | | |||
Multiply the LAWE through by <math>~4\pi x dr</math>, and integrate over the volume of the configuration gives the, | |||
<div align="center"> | |||
<font color="#770000">'''Governing Variational Relation</font><br /> | |||
<table border="0" cellpadding="5" align="center"> | |||
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<math>~0</math> | |||
</td> | |||
<td align="center"> | |||
<math>~=</math> | |||
</td> | |||
<td align="left"> | |||
<math>~ | |||
\int_0^R 4\pi r^4 \gamma P \biggl(\frac{dx}{dr}\biggr)^2 dr | |||
- \int_0^R 4\pi (3\gamma - 4) r^3 x^2 \biggl( \frac{dP}{dr} \biggr) dr | |||
</math> | |||
</td> | |||
</tr> | |||
<tr> | |||
<td align="right"> | |||
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</td> | |||
<td align="center"> | |||
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</td> | |||
<td align="left"> | |||
<math>~ | |||
- 4\pi \biggr[r^4 \gamma Px \biggl(\frac{dx}{dr}\biggr) \biggr]_0^R | |||
- \int_0^R 4\pi \omega^2 \rho r^4 x^2 dr \, . | |||
</math> | |||
</td> | |||
</tr> | |||
<tr> | |||
<td align="right"> | |||
| |||
</td> | |||
<td align="center"> | |||
<math>~=</math> | |||
</td> | |||
<td align="left"> | |||
<math>~ | |||
\int_0^R x^2 \biggl(\frac{d\ln x}{d\ln r}\biggr)^2 \gamma 4\pi r^2P dr | |||
- \int_0^R (3\gamma - 4)x^2 \biggl( - \frac{GM_r}{r} \biggr) 4\pi \rho r^2 dr | |||
</math> | |||
</td> | |||
</tr> | |||
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<td align="right"> | |||
| |||
</td> | |||
<td align="center"> | |||
| |||
</td> | |||
<td align="left"> | |||
<math>~ | |||
+ \biggr[\gamma 4\pi r^3 Px^2 \biggl(-\frac{d\ln x}{d\ln r}\biggr) \biggr]_0^R | |||
- \int_0^R 4\pi \omega^2 \rho r^4 x^2 dr \, . | |||
</math> | |||
</td> | |||
</tr> | |||
</table> | |||
</div> | |||
Now, by setting <math>~(d\ln x/d\ln r)_{r=R} = -3</math>, we can ensure that the pressure fluctuation is zero and, hence, <math>~P = P_e</math> at the surface, in which case this relation becomes, | |||
<div align="center"> | |||
<table border="0" cellpadding="5" align="center"> | |||
<tr> | |||
<td align="right"> | |||
<math>~\omega^2</math> | |||
</td> | |||
<td align="center"> | |||
<math>~=</math> | |||
</td> | |||
<td align="left"> | |||
<math>~ | |||
\frac{\gamma (\gamma -1) \int_0^R x^2 \bigl(\frac{d\ln x}{d\ln r}\bigr)^2 dU_\mathrm{int} | |||
- \int_0^R (3\gamma - 4)x^2 dW_\mathrm{grav} | |||
+ 3^2 \gamma x^2 P_eV}{ \int_0^R x^2 r^2 dM_r} | |||
</math> | |||
</td> | |||
</tr> | |||
</table> | |||
</div> | |||
|- | |||
! style="text-align:center;" width="50%" |<b>Approximation: Homologous Expansion/Contraction</b> | |||
|- | |||
! style="vertical-align:top; text-align:left;" | | |||
If we ''guess'' that radial oscillations about the equilibrium state involve purely homologous expansion/contraction, then the radial-displacement eigenfunction is, <math>~x</math> = constant, and the governing variational relation gives, | |||
<div align="center"> | |||
<table border="0" cellpadding="5" align="center"> | |||
<tr> | |||
<td align="right"> | |||
<math>~\omega^2 \int_0^R r^2 dM_r</math> | |||
</td> | |||
<td align="center"> | |||
<math>~\approx</math> | |||
</td> | |||
<td align="left"> | |||
<math>~ | |||
(4- 3\gamma) W_\mathrm{grav}+ 3^2 \gamma P_eV \, . | |||
</math> | |||
</td> | |||
</tr> | |||
</table> | |||
</div> | |||
|} | |} |
Revision as of 01:25, 19 June 2017
Spherically Symmetric Configurations Synopsis
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New Table Construction
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Equilibrium Structure | |||||||||||||||||||
Detailed Force Balance | Free-Energy Analysis | ||||||||||||||||||
Given a barotropic equation of state, <math>~P(\rho)</math>, solve the equation of
for the radial density distribution, <math>~\rho(r)</math>. |
The Free-Energy is,
Therefore, also,
Equilibrium configurations exist at extrema of the free-energy function, that is, they are identified by setting <math>~d\mathfrak{G}/dR = 0</math>. Hence, equilibria are defined by the condition,
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Virial Equilibrium | |||||||||||||||||||
Multiply the hydrostatic-balance equation through by <math>~rdV</math> and integrate over the volume:
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Stability Analysis | |||||||||||||||||||
Perturbation Theory | Free-Energy Analysis | ||||||||||||||||||
Given the radial profile of the density and pressure in the equilibrium configuration, solve the eigenvalue problem defined by the, LAWE: Linear Adiabatic Wave (or Radial Pulsation) Equation
to find one or more radially dependent, radial-displacement eigenvectors, <math>~x \equiv \delta r/r</math>, along with (the square of) the corresponding oscillation eigenfrequency, <math>~\omega^2</math>. |
The second derivative of the free-energy function is,
Evaluating this second derivative for an equilibrium configuration — that is by calling upon the (virial) equilibrium condition to set the value of the internal energy — we have,
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Variational Principle | |||||||||||||||||||
Multiply the LAWE through by <math>~4\pi x dr</math>, and integrate over the volume of the configuration gives the, Governing Variational Relation
Now, by setting <math>~(d\ln x/d\ln r)_{r=R} = -3</math>, we can ensure that the pressure fluctuation is zero and, hence, <math>~P = P_e</math> at the surface, in which case this relation becomes,
| |||||||||||||||||||
Approximation: Homologous Expansion/Contraction | |||||||||||||||||||
If we guess that radial oscillations about the equilibrium state involve purely homologous expansion/contraction, then the radial-displacement eigenfunction is, <math>~x</math> = constant, and the governing variational relation gives,
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Old Table Construction
Spherically Symmetric Configurations that undergo Adiabatic Compression/Expansion — adiabatic index, <math>~\gamma</math> |
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Equilibrium Structure |
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Detailed Force Balance |
Free-Energy Analysis |
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The Free-Energy is,
Therefore, also,
Equilibrium configurations exist at extrema of the free-energy function, that is, they are identified by setting <math>~d\mathfrak{G}/dR = 0</math>. Hence, equilibria are defined by the condition,
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Virial Equilibrium | |||||||||||||||||||
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Stability Analysis |
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Perturbation Theory |
Free-Energy Analysis |
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Given the radial profile of the density and pressure in the equilibrium configuration, solve the eigenvalue problem defined by the, LAWE: Linear Adiabatic Wave (or Radial Pulsation) Equation
to find one or more radially dependent, radial-displacement eigenvectors, <math>~x \equiv \delta r/r</math>, along with (the square of) the corresponding oscillation eigenfrequency, <math>~\omega^2</math>. |
The second derivative of the free-energy function is,
Evaluating this second derivative for an equilibrium configuration — that is by calling upon the (virial) equilibrium condition to set the value of the internal energy — we have,
|
||||||||||||||||||
Variational Principle |
|||||||||||||||||||
Multiply the LAWE through by <math>~4\pi x dr</math>, and integrate over the volume of the configuration gives the, Governing Variational Relation
Now, by setting <math>~(d\ln x/d\ln r)_{r=R} = -3</math>, we can ensure that the pressure fluctuation is zero and, hence, <math>~P = P_e</math> at the surface, in which case this relation becomes,
|
|||||||||||||||||||
Approximation: Homologous Expansion/Contraction |
|||||||||||||||||||
If we guess that radial oscillations about the equilibrium state involve purely homologous expansion/contraction, then the radial-displacement eigenfunction is, <math>~x</math> = constant, and the governing variational relation gives,
|
See Also
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